We first consider single compact quantum groups to fix the notation and to collect necessary facts on them.
To the best of our knowledge, no analytical attempts have been made to spherical unitary representations (rather than spherical functions) for (Gelfand) pairs \(G < G\times G\)  (via the diagonal embedding) even in the compact quantum group case.
The discussions on compact quantum groups below entirely follows Neshveyev–Tuset's book [1]} (a standard reference on the operator algebra side), but we sometimes use slightly different symbols, such as `\(\mathrm {id}\) ' (instead of `\(\iota \) ' there) denoting the identity map, from their book in order to keep the notation so far.
